Abstract
We study a first-price auction with two bidders where one bidder is characterized by a constant relative risk aversion utility function (i.e., a concave power function) while the other has a general concave utility function. We establish the existence and uniqueness of the optimal strategic markups and analyze the effects of one bidder’s risk aversion level on the optimal strategic markups of him and his opponent’s, the allocative efficiency of the auction, and the seller’s expected revenue, respectively.
Highlights
MM follows the VUS model and extends it to the case of two risk-averse bidders who had the same parametric family of (CRRA) functions, but their measures of risk aversion differ. en, they derive the explicit expressions of asymmetric bidding equilibrium
We generalize the work by MM to a more general case where one bidder is characterized by a constant relative risk aversion (CRRA) utility function while the other has a general concave utility function
We find that when one bidder becomes more risk averse and the other bidder’s risk averse remain unchanged, both bidders reduce their optimal strategic markups, and an illustrative example shows that the rate at which his markup decreases is higher than the rate at which his opponent’s markup decreases; (iii) the allocative efficiency becomes complicated as one bidder becomes more risk averse; (iv) we study effects of one bidder’s risk aversion on expected revenue
Summary
(i) e optimal strategic markups for both bidders (bα, bβ) are characterized by (when uβ(x) xρβ , we get the same optimal strategic markups as in MM). For δ ∈ (0, 0.55), where a 1, ρα 0.9, and ρβ 0.6 It shows that both optimal strategic markups are declining with δ, implying that as bidder β becomes more risk averse, both bα and bβ decrease (i.e., part (i) of Proposition 3 holds), bβ is more rapidly decreased than bα, and the magnitude of the asymmetry effects on the optimal bid markup becomes smaller for bidder α but becomes bigger for bidder β. It shows that both optimal strategic markups are declining with δ, implying that as bidder α becomes more risk averse, both bα and bβ decrease (i.e., part (i) of Proposition 3 holds), bα is more rapidly decreased than bβ, and the magnitude of the asymmetry effects on the optimal bid markup becomes smaller for bidder β but becomes bigger for bidder α. From Proposition 3 and Proposition 4, we can find that the two bidders make uniformly higher bids as bidder β or bidder α becomes more risk averse. is is consistent with the traditional result on symmetric auctions
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